Hillary Clinton is counterpunching just as soon as Republicans strike during their party convention in Cleveland this week, but the former first lady’s campaign largely is limiting its fire to politicians and has shied away from criticism of celebrities and Donald Trump’s children.

The Clinton campaign’s rapid response effort — dubbed “Better Than This” — has inundated reporters with emails throughout the GOP gathering so far, taking relentless aim at Mr. Trump, New Jersey Gov. Chris Christie, former New York City Mayor Rudolph W. Giuliani, retired neurosurgeon Ben Carson and others who have spoken in Cleveland to bash Mrs. Clinton and other Democrats.

The initiative is part of the Clinton campaign’s broader plan to steal at least some of the spotlight during the Republican convention and not let Mr. Trump and other Republicans have the stage all to themselves this week. On Tuesday night, the campaign released a lengthy statement that attacked virtually every speaker who addressed the convention that day, including high-profile Republicans such as Mr. Trump and Mr. Christie but also lesser-known politicians at the state and local level.

One such target was South Carolina Lt. Gov. Henry McMaster.

“McMaster is such a fan of bipartisanship he once stated that ‘we hunt Democrats with dogs down here,’” the Clinton campaign said. “That comment was really just a distraction from McMaster’s true record of bringing Americans together, like the time he joined a country club with a history of discriminating against African Americans, or the time he opposed marriage equality, or that time he supported forced ultra sounds for women seeking abortions.”

Former U.S. Attorney General Michael Mukasey, who served during the Bush administration, also was a target. “Mukasey is just as divisive as Trump, having argued that immigrants facing deportation have no right to a competent attorney,” the Clinton campaign said.

There were similar attacks against most other Republicans who have spoken thus far.

But Mrs. Clinton carefully has avoided attacking most of the nonpoliticians who have appeared on stage in Cleveland. Even despite the plagiarism flap around her speech, Melania Trump was free from criticism from the Clinton campaign. Mrs. Clinton also didn’t directly attack the Trump children, though the campaign did call some of Donald Trump Jr.’s claims “lies” in an email to journalists.

Actors Scott Baio and Antonio Sabato Jr. also escaped shots after their appearances.

The Clinton campaign even avoided criticizing Pat Smith, the mother of a U.S. official killed in the 2012 Benghazi attack. The campaign released a statement citing comments from the family of fellow Benghazi victim Ambassador J. Christopher Stevens — quoting his sister, Anne, saying that Mrs. Clinton shouldn’t be blamed for the attack,

But the campaign didn’t directly address Mrs. Smith, despite her saying in the speech that she blames Mrs. Clinton for the death of her son Sean.

There are, however, a few exceptions.

The Clinton campaign criticized UFC President Dana White just as he took the stage in Cleveland, painting him as hostile to women.

“Hard to believe that the leader of the UFC would have made divisive attacks on women and LGBT Americans, and accused a woman who claimed she was a survivor of domestic violence of ‘terrorizing’ her allegedly abusive partner, who had previously plead ‘no contest’ to other domestic violence charges,” the campaign said in an email.

The campaign also took a shot at Sajid Tarar, founder of American Muslims for Trump, for alleged hypocrisy in his support of the billionaire businessman.

“Tarar supports Trump’s Muslim ban, but has admitted that banning immigrants based on their religion violates the constitution. Oops?” the campaign said.